Collision Probabilities for Continuous-Time Systems Without Sampling

Abstract

Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. These objectives motivate the desire for efficient risk estimation that can be embedded in core decision-making tasks such as motion planning. On one hand, Monte-Carlo (MC) and other samplingbased techniques provide accurate solutions for a wide variety of motion models but are cumbersome in the context of continuous optimization. On the other hand, "direct" approximations aim to compute (or upper-bound) the failure probability as a smooth function of the decision variables, and thus are convenient for optimization. However, existing direct approaches fundamentally assume discrete-time dynamics and can perform unpredictably when applied to continuous-time systems ubiquitous in the real world, often manifesting as severe conservatism. State-of-theart attempts to address this within a conventional discretetime framework require additional Gaussianity approximations that ultimately produce inconsistency of their own. In this paper we take a fundamentally different approach, deriving a risk approximation framework directly in continuous time and producing a lightweight estimate that actually converges as the underlying discretization is refined. Our approximation is shown to significantly outperform state-of-the-art techniques in replicating the MC estimate while maintaining the functional and computational benefits of a direct method. This enables robust, risk-aware, continuous motion-planning for a broad class of nonlinear, partially-observable systems.

Data-Driven Chance Constrained Control using Kernel Distribution Embeddings

“…The most common formulation enforces pointwise chance constraints that ensure the independent satisfaction of each constraint at each time step with high probability (Castillo-Lopez et al, 2019;Lew et al, 2020;Hewing et al, 2020;Polymenakos et al, 2020;Khojasteh et al, 2020;Jasour et al, 2021). In contrast, joint chance constraints guarantee trajectory-wise constraints satisfaction with high probability (Blackmore et al, 2011;Frey et al, 2020;Schmerling and Pavone, 2017;Koller et al, 2018;Lew et al, 2022) which is of particular interest whenever all constraints should be satisfied at all times jointly. For instance, a drone transporting a package should always avoid obstacles and reach its destination with high probability over the distribution of possible payloads.…”

Exact and Bounded Collision Probability for Motion Planning under Gaussian Uncertainty

“…In [8], a Gaussian Process (GP) based technique is employed to learn motion patterns (a mapping from states to trajectory derivatives) to predict possible future obstacles trajectories. First-exist times for Brownian motions are extended to continuous nonlinear dynamics to compute collision probabilities in [20]. Axelrod et al [4] focus exclusively on obstacle uncertainty and formalize a notion of shadows, which are the geometric equivalent of confidence intervals for uncertain obstacles.…”

Probabilistic Collision Constraint for Motion Planning in Dynamic Environments

“…Future obstacle trajectories are predicted using a Gaussian Process (GP) based technique that learns the mapping from states to trajectory derivatives in [1]. In [11] the first-exist times for Brownian motions are leveraged to compute collision probabilities. [2] focus exclusively on environment uncertainty and formalize a notion of shadows, a geometric equivalent of confidence intervals for uncertain obstacles.…”